As soaring college costs saddle more students with debt, colleges are increasingly earmarking aid for students who can fall through the cracks of existing programs: those who aren’t from poor families, can’t necessarily maintain a 4.0 GPA or hit a three-pointer at the buzzer.

California started a scholarship program this school year to help such middle-class students whose families earn as much as $150,000 a year. Pennsylvania rolled out the Ready to Succeed Scholarship, offering as much as $2,000 in aid for residents whose...